Random Explorations (Student Mathematical Library, 98)

"The title “Random Explorations” has two meanings. First, a few topics of advanced probability are deeply explored. Second, there is a recurring theme of analyzing a random object by exploring a random path. This book is an outgrowth of lectures by the author in the University of Chicago Research Experiences for Undergraduate (REU) program in 2020. The idea of the course was to expose advanced undergraduates to ideas in probability research. The book begins with Markov chains with an emphasis on transient or killed chains that have finite Green's function. This function, and its inverse called the Laplacian, is discussed next to relate two objects that arise in statistical physics, the loop-erased random walk (LERW) and the uniform spanning tree (UST). A modern approach is used including loop measures and soups. Understanding these approaches as the system size goes to infinity requires a deep understanding of the simple random walk so that is studied next, followed by a look at the infinite LERW and UST. Another model, the Gaussian free field (GFF), is introduced and related to loop measure. The emphasis in the book is on discrete models, but the final chapter gives an introduction to the continuous objects: Brownian motion, Brownian loop measures and soups, Schramm-Loewner evolution (SLE), and the continuous Gaussian free field. A number of exercises scattered throughout the text will help a serious reader gain better understanding of the material." -- Provided by publisher Cover Title page Contents Preface Chapter 1. Markov Chains 1.1. Definition 1.2. Laplacian and harmonic functions 1.3. Markov chain with boundary 1.4. Green’s function 1.5. An alternative formulation 1.6. Continuous time Further Reading Chapter 2. Loop-Erased Random Walk 2.1. Loop erasure 2.2. Loop-erased random walk 2.3. Determinant of the Laplacian 2.4. Laplacian random walk 2.5. Putting the loops back on the path 2.6. Wilson’s algorithm Further Reading Chapter 3. Loop Soups 3.1. Introduction 3.2. Growing loop at a point 3.3. Growing loop configuration in A 3.4. Rooted loop soup 3.5. (Unrooted) random walk loop measure 3.6. Local time and currents 3.7. Negative weights 3.8. Continuous time Further Reading Chapter 4. Random Walk in Z ^{d} 4.1. Introduction 4.2. Local central limit theorem 4.3. Green’s function 4.4. Harmonic functions 4.5. Capacity for d≥3 4.6. Capacity in two dimensions Further reading Chapter 5. LERW and Spanning Trees on Z ^{d} 5.1. LERW in Z ^{d} 5.2. Marginal distributions for UST in Z ^{d} 5.3. Uniform spanning tree (UST) in Z ^{d} 5.4. The dual lattice in Z 2 5.5. The uniform spanning tree (UST) in Z 2 Further Reading Chapter 6. Gaussian Free Field 6.1. Introduction 6.2. Multivariate normal distribution 6.3. Gaussian fields coming from Markov chains 6.4. A Gibbs measure perspective 6.5. One-dimensional GFF 6.6. Square of the field Further reading Chapter 7. Scaling Limits 7.1. The idea of a scaling limit 7.2. Brownian motion 7.3. Conformal invariance in two dimensions 7.4. Brownian loop soup 7.5. Scaling limit for LERW 7.6. Loewner differential equation 7.7. Self-avoiding walk: ¢=0 7.8. Continuous GFF for d=1,2 Further Reading Appendix A. Some Background and Extra Topics A.1. Borel-Cantelli lemma A.2. Second moment method A.3. Compound Poisson process A.4. Negative binomial process A.5. Increasing jump processes A.6. Gamma process A.7. Lévy processes Bibliography Index Back Cover The emphasis of this book is on discrete models, but the final chapter gives an introduction to the continuous objects: Brownian motion, Brownian loop measures and soups, Schramm-Loewner evolution, and the continuous Gaussian free field. A number of exercises scattered throughout the text will help gain a better understanding of the material.